The St. Petersburg gamble and risk

Pursuing a line of thought initiated by Maurice Allais (1979), I consider whether the mean-risk method of decision making introduced by Harry Markowitz (1959) and other resolves Karl Menger's (1934) version of the St. Petersburg paradox. I provide a conditional answer to this question. I demonstrate that given certain plausible assumption about attitudes toward risk, a certain plausible development of the mean-risk method does resolve the paradox. My chief premiss is roughly that in the St. Petersburg gamble the small chances for large prizes create big risks.     

Newcomb's many problems

Newcomb's paradox rests on two arguments one appealing to the principle of maximizing expected utility and one appealing to dominance in order to generate conflicting recommendations in certain kinds of choice situations. In my essay, I argue that the applications of the principle of maximizing expected utility and of the dominance principle are both fallacious and that the specification of the decision problem is too indeterminate to render a verdict between the two options considered. I also show that if Nozick's case for invoking the dominance principle is taken seriously, it leads to contradictions.     

A conditional expected utility model for myopic decision makers

An expected utility model is formulated and axiomatized which allows the decision maker to specify his actions in the form of partial rather than complete contingency plans, and to simultaneously choose goals and actions in end-mean pairs. Both utility and probability are conditioned on selected goals and actions, and both are defined over the same set of events. For adaptive sequential decision problems this symmetrical treatment of utility and probability permits the expected utility criterion function to be directly updated in each decision period via transitional utility assessments in a manner analogous to Bayes' rule for updating probability distributions.This study is a condensed revised version of discussion papers [14] and [15]. Research underlying this paper was supported by National Science Foundation Grants GS-31276X and GS-35682X. Valuable comments by Professors J. S. Chipman, C. Hildreth, L. Hurwicz, I. Richards, and M. K. Richter are gratefully acknowledged.     

Choice under risk and the security factor: An axiomatic model

The particular attention paid by decision makers to the security level ensured by each decision under risk, which is responsible for the certainty effect, can be taken into account by weakening the independence and continuity axioms of expected utility theory. In the resulting model, preferences depend on: (i) the security level, (ii) the expected utility, offered by each decision. Choices are partially determined by security level comparison and completed by the maximization of a function, which express the existing tradeoffs between expected utility and security level, and is, at a given security level, an affine function of the expected utility. In the model, risk neutrality at a given security level implies risk aversion.     

Analyzing the Demand for Deductible Insurance

This paper investigates aspects of insurance demand related to deductible insurance. In particular, an important issue concerning analysis of the optimal deductible level is resolved. A simple sufficient restriction on the pricing of insurance is given which ensures that the second order condition for choosing the expected utility maximizing deductible level is met for any risk averse decision maker. This restriction is stated and its sufficiency is demonstrated using the level of expected indemnification rather than the level of the deductible as the choice variable in the decision model.     

The two faces of independence: betweenness and homotheticity

This paper shows that expected utility belongs to the intersection of models satisfying betweenness and a homotheticity condition for risky choice. Betweenness models can accommodate variable risk attitudes, originally highlighted by the Allais paradox, by restricting indifference curves to be linear while allowing non-parallelism. Homotheticity, in contrast, restricts indifference curves to be parallel while permitting non-linearities, such as those highlighted by inverse-S probability weighting. Data from an experiment indicate that approximately 2/3s of participants satisfied homotheticity. Of this group, about half satisfied the expected utility hypothesis. The other half violated expected utility and therefore betweenness. These results indicate that homothetic preferences, and in particular those violating betweenness, can be empirically relevant for decision making under risk.

     

Consequentialist foundations for expected utility

Behaviour norms are considered for decision trees which allow both objective probabilities and uncertain states of the world with unknown probabilities. Terminal nodes have consequences in a given domain. Behaviour is required to be consistent in subtrees. Consequentialist behaviour, by definition, reveals a consequence choice function independent of the structure of the decision tree. It implies that behaviour reveals a revealed preference ordering satisfying both the independence axiom and a novel form of sure-thing principle. Continuous consequentialist behaviour must be expected utility maximizing. Other plausible assumptions then imply additive utilities, subjective probabilities, and Bayes' rule.     

Individual vs. couple behavior: an experimental investigation of risk preferences

In this article, we elicit both individuals’ and couples’ preferences assuming prospect theory (PT) as a general theoretical framework for decision under risk. Our experimental method, based on certainty equivalents, allows to infer measurements of utility and probability weighting at the individual level and at the couple level. Our main results are twofold. First, risk attitude for couples is compatible with PT and incorporates deviations from expected utility similar to those found in individual decision making. Second, couples’ attitudes towards risk are found to be consistent with a mix of individual attitudes, women being more influent on couples’ preferences at low probability levels.     

Statistical decisions under ambiguity

This article provides unified axiomatic foundations for the most common optimality criteria in statistical decision theory. It considers a decision maker who faces a number of possible models of the world (possibly corresponding to true parameter values). Every model generates objective probabilities, and von Neumann–Morgenstern expected utility applies where these obtain, but no probabilities of models are given. This is the classic problem captured by Wald’s (Statistical decision functions, 1950) device of risk functions. In an Anscombe–Aumann environment, I characterize Bayesianism (as a backdrop), the statistical minimax principle, the Hurwicz criterion, minimax regret, and the “Pareto” preference ordering that rationalizes admissibility. Two interesting findings are that c-independence is not crucial in characterizing the minimax principle and that the axiom which picks minimax regret over maximin utility is von Neumann–Morgenstern independence.     

Expected utility without utility

This paper advances an interpretation of Von Neumann-Morgenstern's expected utility model for preferences over lotteries which does not require the notion of a cardinal utility over prizes and can be phrased entirely in the language of probability. According to it, the expected utility of a lottery can be read as the probability that this lottery outperforms another given independent lottery. The implications of this interpretation for some topics and models in decision theory are considered.     

Decision analysis using lottery-dependent utility

In this article we show how the lottery-dependent expected utility (LDEU) model can be used in decision analysis. The LDEU model is an extension of the classical expected utility (EU) model and yet permits preference patterns that are infeasible in the EU model. We propose a framework for constructing decision trees in a particular way that permits us to use the principle of optimality and thus the divide and conquer strategy for analyzing complex problems using the LDEU model. Our approach may be applicable to some other nonlinear utility models as well. The result is that, if desired, decision analysis can be conducted without assuming the restrictive substitution principle/independence axiom.     

Causality in the logic of decision

In recent years there has been an active debate between proponents of two different models of rational decision. One model is evidential decision theory, which is characterized by the fact that it holds the principle of maximizing expected utility to be appropriate whenever the states are probabilistically independent of the acts. The other model, causal decision theory, holds that the principle of maximizing expected utility is appropriate whenever the states are causally independent of the acts. The proponents of evidential decision theory include Richard Jeffrey and Ellery Eells, who claim that evidential decision theory has significant advantages over causal decision theory. In this paper I discuss the two main advantages which have been claimed for evidential decision theory, and show that in fact evidential decision theory does not possess either of these advantages.     

Comparing risk preferences over financial and environmental lotteries

This paper investigates whether preferences over environmental risks are best modeled using probability-weighted utility functions or can be reasonably approximated by expected utility (EU) or subjective EU models as is typically assumed. I elicit risk attitudes in the financial and environmental domains using multiple-price list experiment. I examine how subjects?? behavioral, attitudinal, and demographic characteristics affect their probability weighting functions first for financial risks, then for oil-spill risks. I find that most subjects tend to overweight extreme positive outcomes relative to expected utility in both the environmental and financial domains. Subjects are more likely to overemphasize low probability, extreme environmental outcomes than low probability, extreme financial outcomes, leading subjects to offer more support for mitigating environmental gambles than financial gambles with the same odds and equivalent outcomes. I conclude that EU models are likely to underestimate subjects?? willingness to pay for environmental cleanup programs or policies with uncertain outcomes.     

A Rule For Updating Ambiguous Beliefs

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E ^c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.     

An experimental evaluation of the descriptive validity of lottery-dependent utility theory

This article compares the performance of the expected utility (EU) and lottery-dependent expected utility (LDEU) models in predicting the actual choices of experimental subjects among risky options. In the process, we present two approaches for calibrating the LDEU model for an individual decision maker. The results indicate that while LDEU exhibits a higher potential for correctly predicting choice, the version of the model calibrated by indifference judgments does not outperform EU. We suggest a functional form for the parametric functions that defines the LDEU model, and discuss ways in which this function can be incorporated into choice-based assessment approaches to improve predictions.This research was supported in part by the Business Associates Fund at the Fuqua School of Business, Duke University.     

The price for information about probabilities and its relation with risk and ambiguity

In this article, ambiguity attitude is measured through the maximum price a decision maker is willing to pay to know the probability of an event. Two problems are examined in which the decision maker faces an act: in one case, buying information implies playing a lottery, while, in the other case, buying information gives also the option to avoid playing the lottery. In both decision settings, relying on the Choquet expected utility model, we study how the decision maker??s risk and ambiguity attitudes affect the reservation price for ambiguity resolution. These effects are analyzed for different levels of ambiguity of the act. Operating instructions for the elicitation of the reservation price for ambiguity resolution in an experimental setting are provided at the end of the article.     

Compatibility of multiple goal programming and the maximize expected utility criterion

Assuming a decision maker accepts the basic axioms of von Neumann-Morgenstern utility theory and is therefore an expected utility maximizer, this paper argues that the domain of the decision variables in a multiobjective program should be altered in order to guarantee that it will be compatible with the maximize expected utility critierion. Stochastic dominance is employed to approximate this new domain, and for a certain class of decision problems it is shown that this approximation is very good.     

Taste uncertainty and status quo effects in consumer choice

We use reference-dependent expected utility theory to develop a model of status quo effects in consumer choice. We hypothesise that, when making their decisions, individuals are uncertain about the utility that will be yielded by their consumption experiences in different ‘taste states’ of the world. If individuals have asymmetric attitudes to gains and losses of utility, the model entails acyclic reference-dependent preferences over consumption bundles. The model explains why status quo effects may vary substantially from one decision context to another and why some such effects may decay as individuals gain market experience.